Comprehensive Examination

1. Students in mathematics (excluding statistics) must satisfy the basic component of the comprehensive examination in two subjects selected from the following list:
   1. Algebra
   2. Analysis
   3. Topology
   4. Continuous applied mathematics (c.a.m.)
   5. Discrete applied mathematics (d.a.m)
   6. Probability
   7. Statistics

The choice of subjects will be made by the student’s supervisor, in consultation with the student.

2. The basic comprehensive component is satisfied by passing the final examinations in the two one-term core courses for each of the selected subjects:

* For analysis, in addition it is required that the student have demonstrated competence in complex analysis at the advanced undergraduate level.

3. In the event of a failure in one or more attempts of the general comprehensive examinations, a student may either be asked to withdraw from the program, or be required to retake one or more of the examinations.
4. In any event, the basic comprehensive component must be met within 20 months of the student’s initial registration in the case of full-time students and within 40 months in the case of part-time students.
5. All final examination papers in all core courses will be kept by the department for 20 months. A student who has passed the final examination of a core course with a grade of B or better, within this time period, may request to have this be counted as a grade of “pass” towards his or her basic comprehensive component.

The above core courses are offered at Carleton University and the University of Ottawa on an alternating basis.

Each final examination is a formal three-hour written examination reflecting the respective syllabus. Graduate students can take a final examination even if they are not registered in the course; these students have to inform the Instructor of the course by November 15, or March 15 of the given academic year if they intend to write an examination in December or in April, respectively.

All comprehensive examinations will be set and graded by examiners from both universities. A grade of “pass” (a grade of B or better) or “fail” will be assigned for each examination. In addition to the final examination grade, each student registered in the course will be given a grade based on his/her overall performance in the course. Such a grade will be the sole responsibility of the professor teaching the course. A passing grade in the course does not ensure a “pass” for the comprehensive requirement, or vice versa.

1. The advanced comprehensive component will be prescribed by the student’s supervisory committee. It will take the form of a special examination (written and/or oral) based on a syllabus prescribed by the student’s supervisory committee and given to the student six months before the examination. At that time, the form of the advanced comprehensive component will also be specified.
2. The advanced comprehensive component must be completed within 20 months of admission to the Ph.D. program in the case of full-time students and within 40 months in the case of part-time students.
3. A student changing his / her area of specialization will be required to complete the advanced comprehensive component in the new area within a time period specified by the student’s new supervisory committee.

1. Students in mathematics with a specialization in statistics, must satisfy a general comprehensive requirement in two subjects selected from the following list:
   1. Analysis
   2. Algebra
   3. Topology
   4. Probability
   5. continuous applied mathematics (c.a.m.)
   6. discrete applied mathematics (d.a.m)

The choice of subjects will be made by the student’s supervisor, in consultation with the student.

2. The general comprehensive requirement is satisfied by passing the final examinations in the two one-term core courses for each of the selected subjects:

* For analysis, in addition it is required that the student have demonstrated competence in complex analysis at the advanced undergraduate level. The examinations will each be set and graded by examiners from both universities. A grade of “pass” or “fail” will be assigned for each examination.

Note: The evaluation procedure for the general comprehensive requirement is independent of the assignment of grades for the related courses. Students need not be registered in the course to write the examination. A passing grade in the course does not ensure a “pass” for the comprehensive requirement, or vice versa.

3. In the event of a failure in one or more attempts of the general comprehensive examinations, a student may be asked to withdraw from the program, or shall be required to retake one or more of the examinations.
4. Students must attempt to satisfy these regulations as soon as possible after their initial registration. In any event, the general comprehensive requirement must be satisfied within 20 months of the student’s initial registration in the case of full-time students and within 38 months in the case of part-time students.
5. All final examination papers in all core courses will be kept by the departments for 20 months, and a student who enters the Ph.D. program after completing one or more of these courses within this time period may request that his/her examination(s) be remarked for the general comprehensive requirement.

1. The area requirement consists of one three-hour, written, closed-book examination in the area of mathematical statistics.
2. There are two examination sessions annually, one in October and one in February. Students are encouraged to attempt the examination early in their programs. In any event, the Area examination must be passed within 18 months of initial registration (36 months for part time students).
3. Students must notify the director of the Institute of their intent to write the examination at least one month in advance.
4. The Area examination is prepared and evaluated by at least two members of the Institute, appointed by the Director in consultation with the appropriate advisory committee of the Institute.

1. Students in mathematics with a specialization in statistics, must satisfy a general comprehensive requirement in one subject other than their area of specialization, selected from the following list:
   1. Analysis
   2. Algebra
   3. Topology
   4. Probability
   5. Continuous applied mathematics (c.a.m.)
   6. Discrete applied mathematics (d.a.m)

The choice of subjects will be made by the student’s supervisor, in consultation with the student.

2. The general comprehensive requirement is satisfied by passing the final examinations in the two one-term core courses for the selected subject:

*For analysis, in addition it is required that the student have demonstrated competence incomplex analysis at the advanced undergraduate level. The examinations will each be set and graded by examiners from both universities. A grade of “pass” or “fail” will be assigned for each examination.

Note: The evaluation procedure for the general comprehensive requirement is independent of the assignment of grades for the related courses. Students need not be registered in the course to write the examination. A passing grade in the course does not ensure a “pass” for the comprehensive requirement, or vice versa.

3. In the event of a failure in one or more attempts of the general comprehensive examinations, a student may be asked to withdraw from the program, or shall be required to retake one or more of the examinations.
4. Students must attempt to satisfy these regulations as soon as possible after their initial registration.

In any event, the general comprehensive requirement must be satisfied within 20 months of the student’s initial registration in the case of full-time students and within 38 months in the case of part-time students.

5. All final examination papers in all core courses will be kept by the departments for 20 months, and a student who enters the Ph.D. program after completing one or more of these courses within this time period may request that his/her examination(s) be remarked for the general comprehensive requirement.

1. The area requirement consists of two three-hour, written, closed-book examinations, one in mathematical statistics and one in applied statistics.
2. There are two examination sessions annually, one in October and one in February.Students are encouraged to attempt the examination early in their programs.

In any event, the Area examination must be passed within 18 months of initial registration (36 months for part-time students).

3. Students must notify the Director of the Institute of their intent to write the examination at least one month in advance.
4. The Area examination is prepared and evaluated by at least two members of the Institute, appointed by the Director in consultation with the appropriate advisory committee of the Institute.

1. The Special examination consists of an examination related to the student’s proposed thesis.
2. The format of the examination is decided by the student’s advisory committee, in consultation with the student. The examination is scheduled, prepared and evaluated by this committee.
3. The student should request a list of topics for the examination from his advisory committee at least 6 months before the examination. In case of failure, the examination may be repeated once, subject to approval of the student’s advisory committee.

In any event, the Special examination must be passed within 2 years of initial registration (4 years for part-time students).

Statistical Inference (Mathematical Statistics I and II):

• G. Casella and R.L. Berger, Statistical Inference, 2nd ed. Duxbury (or Brooks/Cole Cengage Learning), 2002.
• P.J. Bickel and K.A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Vol. I, 2nd ed., Prentice Hall, 2001.
• G.G. Roussas, A First Course in Mathematical Statistics, Addison-Wesley, 1973.
• E.L. Lehmann and J.P. Romano, Testing Statistical Hypotheses, 3d ed., Springer, 2005.

Casella and Berger:

Chapter 5 (5.4-5.5), Chapter 6 (6.1-6.3), Chapter 7, Chapter 8, Chapter 9 (9.1-9.2), Chapter 10

Bickel and Doksum:

Chapter 1 (1.1-1.6), Chapter 2 (2.1-2.3), Chapter 3 (3.1-3.3), Chapter 4 (4.1-4.5 and 4.9), Chapter 5 (5.1-5.4)

Roussas:

Chapter 8, Chapter 10 (10.1 only), Chapter 11, Chapter 12, Chapter 13, Chapter 15 (15.1-15.4 only), Chapter 16 (16.1-16.2)

E.L. Lehmann and J.P. Romano:

Chapter 1 (1.1-1.9), Chapter 3 (3.1-3.9), Chapter 4 (4.1-4.2, 4.4-4.7, 4.9)

Multivariate Analysis:

• Inge Koch, Analysis of Multivariate and High-Dimensional Data, 2014, Cambridge University Press.
  

• Theodore W. Anderson, An Introduction to Multivariate Statistical Analysis, 3rd Edition, 2003, Wiley.
• Y.L., Tong, The Multivariate Normal Distribution, 1990, Springer.
• R.A. Johnson and D.W. Wichern, Applied Multivariate Statistical Analysis, 6th Edition, 2008, Pearson

Linear Models:

• C.E. McCulloch, S.R. Searle and J.M. Neuhaus, Generalized, Linear, and Mixed Models; 2008, Wiley
• Ronald Christensen, Plane Answers to Complex Questions, 2002, Springer
• C.R. Rao, Linear Statistical Inference and Its Applications, 1973, Wiley

• W.M. John Statistical Design and Analysis of Experiments, MacMillan, 1971.
• Kempthorne Design of Experiments, John Wiley, 1952.
• Bishop, Fienberg and HollandDiscrete Multivariate Analysis, M.I.T. Press, 1975.
• J. Brockwell and R.A. Davis Time Series: Theory and Methods, Springer Series in Statistics, 1987.
• G. Cochran Sampling Techniques, 3rd. edition, Wiley, 1977.
• Lawless Statistical Models and Methods for Lifetime Data, Wiley, 1982.
• STAT 5502 Lecture Notes by J.N.K. Rao
• STAT 5505 Lecture Notes by S. Mills
• STAT 5602 Lecture Notes by J.N.K. Rao

1. Experimental Design
   • STAT 5505 Lecture Notes by S. Mills
   • W.M. John
     • Chapter 2
     • Chapter 3
     • Chapter 4 (omitting 4.8-4.9)
     • Chapter 5 (omitting 5.9)
     • Chapter 6
     • Chapter 7 (omitting 7.8-7.13)
     • Chapter 8 (omitting 8.5-8.6, 8.9-8.16)
     • Chapter 11 (omitting 11.15)
     • Chapter 12 (omitting 12.3-12.5, 12.9-12.10)
     • Chapter 13 (omitting 13.3, 13.5-13.7, 13.9)
   • O. Kempthorne
     • Chapter 8: 8.2, 8.3

Topics: Review of matrix theory and linear model theory; Randomization theory for completely randomized and randomized block designs; analysis with treatment errors; efficiency of randomized block design; Orthogonality of factors; randomization theory for Latin squares and mutually orthogonal latin squares; cross over designs, split plot designs; Combinatorial properties of balanced incomplete block designs; intra and interblock analysis for BIB; group
divisible designs and partially balanced incomplete block designs; Factorial experiments; confounding and fractional replication.

2.  Categorical Data Analysis

1. • STAT 5602 Lecture Notes by J.N.K. Rao
   • Bishop, Fienberg, Holland
     • Chapter 2: 2.4-2.5
     • Chapter 3: 3.3-3.5
     • Chapter 4: 4.2, 4.4
     • Chapter 8: 8.2
     • Chapter 10: 10.6
     • Chapter 11: 11.3
     • Chapter 13: 13.4
     • Chapter 14: 14.3, 14.6-14.9

Topics: Asymptotic theory for chi-squared and G2. Maximum likelihood estimation for exponential families, two fundamental equivalence theorems, Haberman’s approach, ordinal data (two-way tables), multiarray tables and loglinear models, model selection; weighted least squares approach, tests of symmetry and quasisymmetry; logic models; measures of association.

3.  Time Series 

1. • Brockwell & Davis
     • Chapter 3
     • Chapter 4: 4.1-4.4
     • Chapter 5
     • Chapter 7
     • Chapter 8

4. Reliability Theory
   • Lawless
     • Chapter 1: 1.4
     • Chapter 2: 2.3-2.4
     • Chapter 6: 6.1-6.5; 6.7
     • Chapter 7: 7.2-7.4
     • Chapter 8: 8.2

Topics: Nonparametric estimation of survival function; parametric models and maximum likelihood estimation; exponential and Weibull regression models; nonparametric hazard function models and associated inference; rank tests with chi-squared data.

5.   Sampling Theory and Methods

1. • STAT 5502 Lecture notes by J.N.K. Rao
   • Cochran
     • Chapter 5A
     • Chapter 6
     • Chapter 9: 9A.1-9A.4; 9A.6-9A.11; 9A.13
     • Chapter 10: 10.4; 10.6; 10.9-10.10
     • Chapter 11: 11.6-11.10; 11.13-11.14; 11.17-11.20
     • Chapter 12: 12.2-12.13

Topics: Unified theory for standard errors; unequal probability sampling with and without replacement, stratification; Ratio estimation; domain estimation; post stratification, multistage sampling unified theory; nonlinear statistics: jacknife, balanced repeated replication, linearization; two-phase sampling; nonresponse and importation; response array.

6.   Modern Computational/Applied Statistics

1. • References:
     • [1] Efron, B., and Tibshirani, R. “An Introduction to the Bootstrap”, Chapman and Hall, 1993.
     • [2] Hastie, T. and Tibshirani, R. “Generalized additive models”, Chapman and Hall, 1990.
     • [3] Ripley, B. “Pattern Recognition and Neural Networks”, Cambridge University Press, 1996.
   • Lecture notes relevant to:
     • Chapters 7, 8, 9, 10, 11, 12, 13, 14, 22 of [1]
     • Chapter 2 of [2]
     • Chapters 5, 6, 7 of [3]

Topics: Resampling and computer intensive methods: bootstrap, jackknife, and data-splitting methods with applications to bias estimation, variance estimation, confidence intervals, and regression analysis; smoothing methods in curve estimation; statistical classification and pattern recognition: error counting methods, optimal classifiers, combined classifiers, use of bootstrap in estimating the bias of the misclassification error rate, nearest neighbour classifiers, neural network classifiers, and tree classifiers.